#### Which of the following is not a quadratic equation?$\\(A)2(x-1)^{2}=4x^{2}-2x+1\\ (B)2x-x^{2}=x^{2}+5\\ (C)(\sqrt{2}x+\sqrt{3})^{2}+x^{2}=3x^{2}-5x\\ (D)(x^{2}+2x)^{2}=x^{4}+3+4x^{3}$

$(C)(\sqrt{2}x+\sqrt{3})^{2}+x^{2}=3x^{2}-5x$

Quadratic equation: A quadratic equation in x is an equation that can be written in the standard form $ax^{2} + bx + c = 0$, $a \neq 0$

Where a, b and c are real numbers

$2(x-1)^{2}=4x^{2}-2x+1$

$2(x^{2}+1-2x)=4x^{2}-2x+1$

[using (a – b)2 = a2 + b2 – 2ab]
$\\2x^{2}+2-4x-4x^{2}+2x-1=0\\ -2x^{2}-2x+1=0$

It is a quadratic equation because it is in the form

$ax^{2} + bx + c = 0$

(B)

$\\2x-x^{2}=x^{2}+5\\ 2x-x^{2}-x^{2}-5=0\\ -2x^{2}+2x-5=0\\ -2x^{2}+2x-5=0$

It is a quadratic equation because it is in the form $ax^{2} + bx + c = 0$

(C)
$(\sqrt{2}x+\sqrt{3})^{3}+x^{2}=3x^{2}-5x\\ (\sqrt{2}x)^{2}+(\sqrt{3})^{2}+2(\sqrt{2}x)(\sqrt{3})+x^{2}=x^{2}-5x$      $\left [using \;(a+b)^{2}=a^{2}+b^{2}+2ab \right ]$

$2x^{2}+3+2\sqrt{6}x+x^{2}-3x^{2}+5x=0\\ (2\sqrt{6}+5)x+3=0$

It is not in the form of $ax^{2} + bx + c = 0$

Hence it is not a quadratic equation.

(D)
$(x^{2} + 2x)^{2} = x^{4} + 3 + 4x^{3}\\ (x^{2})^{2} + (2x)^{2} + 2(x^{2})(2x) = x^{4} + 3 + 4x^{3}$     $\left [using \;(a+b)^{2}=a^{2}+b^{2}+2ab \right ]$
$x^{4} + 4x^{2} + 4x^{3} - x^{4} - 3 - 4x^{3} = 0\\ 4x^{2}-3=0$

It is in the form of $ax^{2} + bx + c = 0$

Hence it is a quadratic equation

Only option (C) is not in the form of $ax^{2} + bx + c = 0$

Hence (C) is not a quadratic equation.